\chapter{Preliminary experimental setup} % (fold)
\label{cha:early_setup}

I have previously reported results demonstrating all-optical switching with transverse optical patterns in \cite{Dawes_2005aa} and \cite{Dawes_2008aa}. The earlier of these, published in 2005, included data that was collected using the first version of my experimental setup for all-optical switching based on patterns. The second, published in 2008, includes a more detailed presentation of the results, and describes some improvements to the experimental system. Finally, the results presented in this thesis reflect additional improvements to the experimental system. The purpose of this Appendix is to outline the various modifications that have been made to my experimental system, and describe the resulting improvements in the operation of my all-optical switch.

\section{Vapor cell} % (fold)
\label{sec:vapor_cell}

There are two different vapor cells that I have used. The first cell, used in the preliminary data collection, contains isotopically-enriched rubidium vapor ($>$ 90\% $^{87}$Rb) in a 5-cm-long glass cell heated to 67 $^{\circ}$C (corresponding to an atomic number density of $\sim 2 \times 10^{11}$ atoms/cm$^3$). The cell is made entirely of pyrex glass and the entrance and exit windows are roughly parallel; thus the entire cell must be tilted with respect to the incident laser beams to prevent possible oscillation between the uncoated windows.

The pyrex windows introduce enough aberration in the pump beam wavefronts that they provide pattern selection via symmetry-breaking. In the ideal situation, the only symmetry-breaking is provided by the pump-beam misalignment, or an other source that can be carefully controlled. However, the pyrex windows limited the performance of the system by coupling the symmetry-breaking due to the pump-beam alignment with symmetry-breaking due to cell-window aberrations. With the pyrex cell, I found that the position of the cell windows within the path of the pump beams had a significant effect on the orientation and symmetry of the generated patterns. The pyrex cell was sufficient to demonstrate the extreme sensitivity of the generated patterns to a weak switch-beam, yet the ultimate performance of the switch was limited by the amount of symmetry breaking introduced by the vapor cell windows.

To overcome this problem, I now use a vapor cell constructed with uncoated quartz windows installed at opposite 11$^\circ$ angles to prevent oscillations between the entrance and exit surfaces. The cell was purchased from Triad Technology Inc., part no. TT-RB-50-V-Q. Like the pyrex cell, this quartz cell contains no buffer gas and is not paraffin coated. With this cell, I observe highly symmetric generated patterns for all pump-beam alignments with $\theta_p<0.1$~mrad. The position of the transmitted beam on the cell windows does not substantially affect the symmetry of the generated patterns, and even tilting the vapor cell while observing the patterns does not have a significant effect on their symmetry.

Another difference between the pyrex vapor cell and the quartz cell is that the quartz cell contains rubidium vapor with naturally abundant isotopes ($\sim$ 72\% $^{85}$Rb, 28\% $^{87}$Rb). My preliminary switching experiments were conducted near the D$_2$ transition in $^{87}$Rb, ($^5\text{S}_{1/2} \rightarrow {^5\text{P}_{3/2}}$, 780~nm wavelength), but I observe instability generated light near either the $D_1$ or the $D_2$ transition and in either isotope. The threshold for the instability is lowest for the $D_2$ transition where I also find that the switching is most sensitive. The pyrex cell contained isotopically-enriched $^{87}$Rb because it was previously used in a different experiment and was readily available for my preliminary work. The presence of $^{85}$Rb in the quartz cell does not seem to have any detrimental effects on the generation of optical patterns or the switch response: my results from experiments using the quartz cell show a factor of five reduction in the number of photons required to actuate the switch.

% section vapor_cell (end)

\section{Magnetic shielding} % (fold)
\label{sec:magnetic_shielding}

My preliminary data was collected using a vapor cell placed within a Helmholtz coil that served to cancel the component of the Earth's magnetic field along the pump-beam axis. The primary reason for this was to reduce the amount of forward-pump power rotated into the vertical polarization by the resonant Faraday effect. The instability-generated light is detected by separating it from the pump beams via polarizing beamsplitter; hence, pump-light that has its state of polarization rotated within the medium will appear as additional background and makes detection of the generated light difficult.

After the pyrex vapor cell was replaced with the quartz cell, I observed patterns that would spontaneously rotate when the pump-beams were well-aligned. Similar spontaneous pattern rotation has been observed in a single-mirror feedback system where weak transverse magnetic fields are found to induce pattern rotation \cite{Huneus_2003aa}. The single-mirror feedback system differs from my system in that a single forward pump wave propagates through the nonlinear medium and is reflected to propagate back through the medium in the opposite direction. In one sense, this is still a counterpropagating beam system, however the feedback mirror couples the generated light back into the system after an amount of free-space diffraction that is controlled by the distance between the mirror and the nonlinear medium. Hexagonal pattern formation has also been observed in the single-mirror system and there are many qualitative similarities between the two-pump system I use, and the single-mirror feedback system. For this reason, I used the results from single-mirror-feedback experiments to guide my attempts to stabilize the spontaneous pattern rotation. In light of the result that weak transverse magnetic fields can induce pattern rotation, I chose to shield the cell from external magnetic fields.

To eliminate transverse and longitudinal magnetic fields, I place the vapor cell within a cylinder constructed from high-permeability $mu$-metal, M$\mu$Shield Inc. custom order rolled and welded sheet constructed from 0.025'' thick high permeability material. The shield has an inner diameter of 5 cm and is 20.5 cm long. The $mu$-metal was annealed in a hydrogen environment after construction in order to increase the permeability and thus increase the magnetic field attenuation. I have measured field attenuation of $>10^4$ in the central third of the shield, corresponding to the cell location. In fact, the attenuation of external fields is $>10^2$ for all but the last 3 cm at either end of the cell.

Using this magnetic shield, the unperturbed pattern orientation is stable for several days and depends solely on the pump beam alignment and intensity.

% section magnetic_shielding (end)

\section{Pump beam symmetry} % (fold)
\label{sec:pump_beam_symmetry}

As the two previous sections illustrate, the primary challenge in optimizing my experimental system for sensitive all-optical switching is the elimination of extraneous symmetry-breaking. To eliminate as much asymmetry as possible, the final experimental modification I have made to my experimental apparatus is the replacement of the single-mode optical fiber that provides spatial filtering of the pump beams. In my preliminary work, I used an optical fiber that was polished such that the entrance and exit face was 8$^\circ$ from normal incidence. This polish standard, known as APC, is one of two options, the other being flat-polished or PC. The benefit of using fibers with APC connectors at each end is the suppression of back-reflections and Fabry-Perot fringes due to the cavity formed by the entrance and exit face of the fiber. One drawback to the APC fiber connector is that unless a custom collimation lens is used, the fiber output will not be a round gaussian mode, the intensity distribution will have a slight amount of ellipticity.

Pump beam ellipticity is a source of symmetry breaking in the system, and can provide pattern selection or lead to pinned patterns. Using the APC fiber output coupler, the pump beam vertical and horizontal waists differed by $\sim$10\%. By replacing the APC single mode fiber with one that had on APC end and one PC end (ThorLabs Inc., custom fiber patch cable 780HP with one end FC/APC and one end FC/PC), I reduced the pump-beam ellipticity to $<0.5\%$, below the precision of the measurement.\footnote{The beam profiles were measured using both a 10-$\mu$m pinhole mounted on a translation stage, and a calibrated CCD beam profiler. The CCD pixels are also $10\mu$m, hence the limited precision of either measurement.} 

% section pump_beam_symmetry (end)

\section{Modulational instability} % (fold)
\label{sec:modulational_instability}

The modulational instability (MI) that I observe in my experimental system contributes to the sensitivity of the switch by destabilizing the pattern orientation. However, if the amplitude of the MI is large, the data fidelity, and the sensitivity begin to decrease. I find there is an optimum MI amplitude that allows high contrast between the two switch states, but also provides high sensitivity by inducing intensity fluctuations having a period that matches the characteristic pattern-rotation time.

I have found two separate ways to control the MI. Varying the total pump power affects the MI amplitude, while increasing or decreasing the pump-beam misalignment $\theta$ serves to increase or decrease the MI frequency, respectively. In my preliminary work, I was not able to continuously vary the pump-beam misalignment without significantly changing the generated pattern because the pyrex cell windows had a large effect on the pattern symmetry. Preliminary switching data, collected using the pyrex vapor cell, is shown in Fig.~\ref{fig:old_vs_new}(a). The data shown were collected with the pump beams 20\% above threshold, hence the MI was large due to the pump beam power being further above threshold.

Comparison with Fig.~\ref{fig:old_vs_new}(b) shows the significant reduction of the MI amplitude after improvements to the experimental system and operating closer to the instability threshold. Furthermore, the frequency of the MI has been optimized to match the response time of the switch and hence leads to more sensitive switch response. Prior to minimizing the modulational instability, the lowest switching photon number I observed was 2,700. By suppressing the modulational instability, and hence lowering the detection threshold for reliable switching, I have observed reliable switching with as few as 600 photons. It should be noted that the response time of the data shown in Fig.~\ref{fig:old_vs_new}(a) and (b) should not be compared directly to one another because the input switch beam power was not the same across these two measurements. For the data shown in Fig.~\ref{fig:old_vs_new}(a) the input power was 3~nW and for the data shown in Fig.~\ref{fig:old_vs_new}(b) the input power was 900~pW.

\begin{figure}[htbp]
  \begin{center}
    \includegraphics[scale=1]{Figures/old_vs_new.pdf}
  \end{center}
  \caption[Preliminary switching data showing large modulational instability amplitude.]{Comparison of modulational instability for preliminary data and recent data. a) Preliminary data shows large modulation amplitude. b) In recent data, the MI amplitude is suppressed by operating close to threshold.}
  \label{fig:old_vs_new}
\end{figure}

% section modulational_instability (end)

% chapter preliminary_experimental_setp (end)